Europ. J. Combinatorics (2002) 23, 719–732 doi:10.1006/eujc.2002.0588 Available online at http://www.idealibrary.com on Bridging Semisymmetric and Half-Arc-Transitive Actions on Graphs DRAGAN MARUSIˇ Cˇ † AND PRIMOZˇ POTOCNIKˇ A generalization of some of Folkman’s constructions (see (1967) J. Comb. Theory, 3, 215–232) of the so-called semisymmetric graphs, that is regular graphs which are edge- but not vertex-transitive, was given by Marusiˇ cˇ and Potocnikˇ (2001, Europ. J. Combinatorics, 22, 333–349) together with a 1 natural connection between graphs admitting 2 -arc-transitive group actions and certain graphs admit- ting semisymmetric group actions. This connection is studied in more detail in this paper. Among others, a sufficient condition for the semisymmetry of the so-called generalized Folkman graphs aris- 1 ing from certain graphs admitting a 2 -arc-transitive group action is given. Furthermore, the concepts of alter-sequence and alter-exponent is introduced and studied in great detail and then used to study the interplay of three classes of graphs: cubic graphs admitting a one-regular group action, the cor- 1 responding line graphs which admit a 2 -arc-transitive action of the same group and the associated generalized Folkman graphs. At the end an open problem is posed, suggesting an in-depth analysis of 1 the structure of tetravalent 2 -arc-transitive graphs with alter-exponent 2. c 2002 Elsevier Science Ltd. All rights reserved. 1. INTRODUCTORY REMARKS A digraph D = (V, A) consists of a finite set of vertices V (D) = V and a set of arcs A(D) = A ⊆ (V × V ) \{(v, v)|v ∈ V }. For an arc (u, v) of a digraph D we say that u and v are the vertices of (u, v), more pecisely, u is the tail and v is the head of (u, v). Also, we say that v is an out-neighbour of u and that u is an in-neighbour of v. The symbols N +(v), N −(v) and N(v) denote the set of out-neighbours of v, the set of in-neighbours of v and the union N +(v) ∪ N −(v), respectively, while the symbols deg+(v) = |N +(v)| and deg−(v) = |N −(v)| denote the out-degree and the in-degree of v, respectively. The minimal out-degree and in-degree of a digraph D are denoted by δ+(D) and δ−(D), respectively. If deg−(v) is constant for all v ∈ V (D), we say that D is in-regular. Similarly, if deg+(v) is constant for all v ∈ V (D), we say that D is out-regular. A digraph D is regular if it is both in- and out-regular. If deg−(v) = deg+(v) for all v ∈ V (D), we say that D is balanced. Note that a regular digraph is always balanced. If for all u, v ∈ V (D), we have that (u, v) ∈ A(D) whenever (v, u) ∈ A(D), we say that D is a graph. An edge of a graph is an unordered pair {u, v} (also denoted by uv) such that (u, v) is an arc of the graph. The set of edges of the graph X is denoted by E(X). We refer the reader to [9, 25, 28] for group-theoretic concepts not defined here. Let X be a graph and G a subgroup of the automorphism group AutX of X. We say that X is G-vertex-transitive, G-edge-transitive and G-arc-transitive if G acts transitively on V (X), 1 E(X) and A(X), respectively. Furthermore, X is said to be (G, 2 )-arc-transitive if it is G- vertex-transitive, G-edge-transitive but not G-arc-transitive. In the special case when G = 1 AutX we say that X is vertex-transitive, edge-transitive, arc-transitive, and 2 -arc-transitive 1 if it is (AutX)-vertex-, (AutX)-edge-, (AutX)-arc transitive, and (AutX, 2 )-arc-transitive, respectively. Let G be a subgroup of AutX such that X is G-edge-transitive but not G-vertex-transitive. Then X is necessarily bipartite, where the two parts of the bipartition are orbits of G. Clearly †Corresponding address: IMFM, Department of Mathematics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia. 0195–6698/02/080719 + 14 $35.00/0 c 2002 Elsevier Science Ltd. All rights reserved. 720 D. Marusiˇ cˇ and P. Potocnikˇ if X is regular then these two parts have equal cardinality. The graph X is said to be G- semisymmetric if it is a regular G-edge- but not G-vertex-transitive graph, and it is said to be semisymmetric in the special case G = AutX. The study of semisymmetric graphs was initiated by Folkman [13] who gave a construction of several infinite families of such graphs. The smallest graph in his construction has 20 vertices and happens to also be the small- est semisymmetric graph. At the end of his paper several open problems were posed, most of which have already been solved (see [2, 3, 10, 11, 15–17]). Folkman’s construction of semisymmetric graphs was generalized by Marusiˇ cˇ and Potocnikˇ in [22], where an interest- 1 ing connection with 2 -arc-transitive group actions was suggested. It is precisely the interplay 1 of semisymmetric and 2 -arc-transitive group actions that constitutes the central topic of this article. Let H be a transitive permutation group acting on a set V and let v ∈ V . There is a 1-1 correspondence between the set of suborbits of H, that is, the set of orbits of the stabilizer Hv on V , and the set of orbitals of H, that is, the set of orbits in the natural action of H on V × V , with the trivial suborbit {v} corresponding to the diagonal {(v, v) : v ∈ V }. For an orbital 0 we let S0,v = {w|(v, w) ∈ 0} denote the suborbit of H (relative to v) associated with 0. Conversely, for a suborbit S of H relative to v we let 0S,v be the associated orbital in the above 1-1 correspondence. The paired orbital 0−1 of an orbital 0 is the orbital {(v, w) : (w, v) ∈ 0}. If 0−1 = 0 we −1 say that 0 is self-paired. Similarly, for a suborbit S of H relative to v we let S = S0−1 , v denote the paired suborbit of S. If S−1 = S we say that S is self-paired. The orbital digraph XE(H, V ; 0) of (H, V ) relative to 0, is the digraph with vertex set V and arc set 0. The underlying undirected graph of XE(H, V ; 0) will be called the orbital graph of (H, V ) relative to 0 and will be denoted by X(H, V ; 0). If 0 = 0−1 is a self-paired orbital then X(H, V ; 0) admits a vertex- and arc-transitive action of H. On the other hand, if 0 is not self-paired then X(H, V ; 0) admits a vertex- and edge- but not arc-transitive action of H, in short, a 1 2 -arc-transitive action of H. For a permutation τ of V contained in the normalizer of the permutation group H in the symmetric group SymV we let 0τ denote the set {(xτ , yτ )|(x, y) ∈ 0}. Since τ normalizes H, the set 0τ is also an orbital of H. If v ∈ V is left fixed by τ and S is the suborbit of H corresponding to the orbital 0 relative to the vertex v, then the set Sτ = {sτ |s ∈ S} is a suborbit of H, which corresponds to the orbital 0τ relative to the vertex v. The following definition of a generalized Folkman graph, introduced in [22], generalizes one of the original Folkman constructions of semisymmetric graphs [13]. DEFINITION 1.1. Let H be a transitive permutation group on a the set V , 0 its orbital, let k ≤ 2 be an integer and let τ be a permutation of V contained in the normalizer of H in SymV k such that τ ∈ H. Let B = {Bx |x ∈ V } and V0 j = {x0 j |x ∈ V, j ∈ Zk} be k +1 copies of the set V . Let Y = Y (H, V, 0, τ, k) denote the graph with vertex set B ∪ V00 ∪ · · · ∪ V0k−1 and τ j edge set {x0 j By|(x, y) ∈ 0 }. The generalized Folkman graph F(H, V, 0, τ, k) is obtained from Y (H, V, 0, τ, k) by expanding each Bx to a k-tuple of vertices x10, x11,..., x1,k−1 each retaining the neighbours’ set of Bx . For j ∈ Zk we let V1 j denote the set {x1 j |x ∈ V }. The generalized Folkman graph F = F(H, V, 0, τ, k) is always G-semisymmetric for some group G ≤ AutF (see [22] for details). The following simple proposition [22, Proposi- tion 1.2] gives a sufficient condition for semisymmetry of a generalized Folkman graph. S PROPOSITION 1.2. If no k distinct vertices in V0 j have the same set of neigh- j∈Zk bours in the graph Y (H, V, 0, τ, k), then the generalized Folkman graph F(H, V, 0, τ, k) is semisymmetric. Semisymmetry 721 The main purpose of this article is to study the natural connection of graphs admitting 1 semisymmetric group actions and graphs admitting 2 -arc-transitive group actions as sug- gested in [22]. In Section 2 we give a sufficient condition for the semisymmetry of gener- 1 alized Folkman graphs arising from certain graphs admitting a 2 -arc-transitive group action (Theorem 2.1).